Operational Mathematics of Set Operations: Extending Iteration Counts from Integers to Complex Numbers and the Unification of Idempotent and Non-Idempotent Logic

We develop a rigorous theory of extending the iteration count of set-theoretic operations---in particular, the binary union and its adjoint ^*---from natural numbers successively to integers, rationals, reals, and complex numbers. A core discovery is that classical idempotent set union forces a collapse of the continuous iteration: all positive-order iterates reduce to the first step, reflecting the deep algebraic constraint of idempotence. To obtain a meaningful, non-degenerate continuous iteration, we introduce a parametric family of logical operations (weighted unions) that break idempotence. For these non-idempotent operations we construct unique real-analytic and complex-analytic iteration semigroups, classify their analytic structure, and prove the strict layering of the associated hyperoperation hierarchy. The classical idempotent case emerges as the boundary limit 1 of this continuous family, exhibiting a Stokes phenomenon at the idempotent point. All constructions are placed on the common ground of a general complete Boolean algebra and its Stone duality embedding into a commutative Banach algebra. The theory is self-contained; every substantial statement is accompanied by a complete proof.